THE STAR FORMATION HISTORY OF THE MILKY WAY'S NUCLEAR STAR CLUSTER^*

In terms of mass, the resolved stellar population represents only the tip of the iceberg. The bulk of the stellar mass is unresolved. Therefore the diffuse background emission of the GC contains valuable information on the cluster composition and its formation. Naturally it is very challenging to estimate the unresolved background in a crowded stellar field. A significant fraction of the background light originates from the uncorrected seeing halos of bright stars. Furthermore, anisoplanatism complicates precise photometry across the FOV when using AO. By using multiple point-spread function (PSF) photometry on NACO images, Schödel (2010) tried to overcome these limitations. The resulting photometry for resolved stars was published in Schödel et al. (2010) and a map containing the unresolved background flux was published in Schödel (2010). We used H-band data from both publications to derive the fraction of light contained in the diffuse background. The H-band has the advantage that it is least affected by the surrounding nebular emission.

Since no completeness map was published by Schödel (2010), we used a superb H-band mosaic with Strehl >20% from the 2010 March 31. We applied StarFinder (Diolaiti et al. 2000) and converted our detections in magnitudes by referencing them to Table A2 of Schödel et al. (2010). To derive the completeness, we applied a common technique of creating and re-detecting artificial stars of various brightness. As a limiting magnitude, separating diffuse background from resolved sources, we used m_{H, cut} = 19.45 ± 0.12. The magnitude was chosen such that the total uncorrected flux of all detected stars is equal to the completeness-corrected flux of stars brighter than the magnitude cut. Using an extinction of A_H = 4.65 ± 0.12 (Fritz et al. 2011) and a GC distance of R₀ = 8.3 ± 0.35 kpc (Ghez et al. 2008; Gillessen et al. 2009) we obtained an absolute separating magnitude of M_{H, cut} = 0.27 ± 0.34. The error estimate accounts for the absolute photometric uncertainty, the cutoff, extinction, and distance uncertainties. We then subtracted the difference between the stellar flux (up to the limiting magnitude) in our H-band image and the stellar flux in the Schödel et al. (2010) list from the diffuse background flux. Furthermore, we subtracted the known early-type stars since they dominate the light in the inner 10''.

We find that the diffuse background (m_H > 19.45 ± 0.12) contains H_diff/H = 27% ± 9% of the total (diffuse + resolved) H-band flux. The 1σ error contains the variation across the field (6%) and the uncertainty due to the cutoff between diffuse and resolved light M_{H, cut} (2%). Taking into account the completeness, we added another 3% systematic error due to undetected early-type stars.

To derive the total H-band luminosity of the inner 1.2 pc, we used the resolved and unresolved stellar H-band flux stated in Schödel et al. (2010) and Schödel (2010). To account for the spatially varying extinction of the GC, we used their published extinction map. Unfortunately, the extinction map is only available for K-band. Therefore we scaled the map to an average H-band extinction of A_H = 4.65 (Fritz et al. 2011). Correcting for extinction and taking the distance into account, we obtained a luminosity of H = 4.51 ± 0.54 × 10⁶ L_{H, ☉} enclosed within a projected radius of r_2D < 1.2 pc. From this we subtracted the contribution of known early-type stars (0.64 × 10⁶ L_{H, ☉}). While the young stars are known to be close in 3D to the center, only 50% ± 5% of the old stars at a projected distance of r_2D < 1.2 pc are also contained within a 3D distance of 1.2 pc (assuming the radial density profile of Schödel et al. 2007). Applying both corrections we find a total luminosity of H(R < 1.2 pc) = 1.94 ± 0.33 × 10⁶ L_{☉, H}. Of this value 0.52 ± 0.2 × 10⁶ L_{☉, H} can be attributed to the diffuse background. The dynamical mass enclosed (excluding the SMBH) is M(R < 1.2 pc) = 1.4 ± 0.7 × 10⁶ M_☉ (Genzel et al. 2010). Thus we derive a total mass-to-light ratio of M/H = 0.7 ± 0.4 M_☉/L_{☉, H} and a diffuse mass-to-light ratio of M/H_diff = 2.6 ± 1.7 M_☉/L_{☉, H}.

By assuming an average intrinsic color of m_{H − K} = 0.2 we can convert the total H-band luminosity into a K-band luminosity of K(R < 1.2 pc) = 3.0 ± 0.44 × 10⁶ L_{☉, K}. The corresponding mass-to-light ratio is M/K = 0.5 ± 0.3 M_☉/L_{☉, K}. The mass-to-diffuse-light ratio is M/K_diff = 1.9 ± 1.2 M_☉/L_{☉, K}. The bolometric mass-to-light ratio is M/L_bol = 0.7 ± 0.4 M_☉/L_{☉, bol}.

Very good seeing conditions and long integration times on two SINFONI fields allowed us to perform the deepest census of the GC population so far. The two fields cover roughly 18 arcsec² and are ∼50% complete down to m_K < 18. The location of the fields can be seen in Figure 1. Both fields were placed in regions with low extinction and without bright stars in the vicinity. One of the fields (East), at a distance of 74, probes the disk region. The second field (North) is located at the outer rim of the disks at 135. The richness of faint stars can be seen in Figure 1. The identification of a m_K < 17 star as late-type requires only two hours of integration on source. The prominent CO bandheads are easy to identify and the multiplicity of the lines helps to rule out misidentifications in noisy spectra. However, the identification of an early-type star is much more challenging. The early-type stars are identified by the hydrogen Brγ line. Unfortunately, the line is less prominent than the CO bandheads. The identification is further hindered by the surrounding nebular Brγ emission. This emission is patchy and varies significantly in intensity and velocity on scales of 1''. Depending on the location of the star, this can lead to a wrong background subtraction during the data analysis mocking a stellar line. Because the stellar absorption line in A-stars is a broad Lorentzian, it is clearly different from the narrow Gaussian emission line of the gaseous background. Thus in case of sufficient signal-to-noise ratio (S/N) an identification is possible. To achieve a sufficient S/N an integration time of about five hours on source is required to identify an early-type star as such with m_K < 18.

Zoom In Zoom Out Reset image size

Figure 1 shows the deep fields. Red and blue circles mark stars identified as late-type or early-type. One star was not confirmed as early-type (green circle), but showed no CO bandheads and is therefore deemed to be an early-type candidate. Within the 18 arcsec² of the two fields, we thus detected five early-type stars and one candidate. Their spectral type was determined by comparison with template spectra of Wallace & Hinkle (1997) and the absolute K-band magnitude M_K taken from Cox (2000). Three of them were brighter than m_K < 17 and were identified as dwarfs with spectral types between B2 and B8. The two faintest early-type stars identified were B9/A0 dwarfs with m_K = 17.2 and m_K = 17.6. These stars are the faintest main-sequence stars reported so far in the GC. Stars of that spectral type have main-sequence lifetimes between 360 and 730 Myr and masses between 2.2 and 2.8 M_☉ (Cox 2000). The spectra of the two A-stars are shown in Figure 3. For comparison, two spectra of late-type giants of equal K-band luminosity are shown. The candidate is of similar luminosity and falls in the same main-sequence category.

Zoom In Zoom Out Reset image size

The deep census of the GC allowed us to construct a KLF with 12 < m_K < 18 (Figure 4). The best-fitting slope of the luminosity distribution dlog N/dm_K = β is found to be β = 0.33 ± 0.03. This is consistent with the results of Buchholz et al. (2009), who determined the slope of stars with a limiting magnitude of m_K < 15.5. As already discussed by Alexander & Sternberg (1999), Genzel et al. (2003), and Buchholz et al. (2009), the slope is typical for an old, bulge-like population. This is in good agreement with our H-R diagram fitting (Section 5.1) showing that most of the star formation occurred more than 5 Gyr ago. However, the KLF alone is not sufficient to constrain the star formation history for an old population. The KLF alone cannot constrain the actual formation history and the IMF since the KLF slope is very insensitive to both parameters (Löckmann et al. 2010). For example, the admixture of young and bright giants is hardly detectable in the KLF due to their low number. Yet they can be easily identified in the H-R diagram.

Zoom In Zoom Out Reset image size

One of the observed fields is centered on the disk region, while the second field probes the outer rim of the disks. Thus the early-type stars in the two observed fields are potential disk members. To test for membership, we used the same method as described in Bartko et al. (2010). The 3D velocity and the positions of all five stars are given in Table 2. The last column indicates a likely disk membership. We find that the two A-stars are clearly not disk members. Among the brighter B-stars, only one is consistent with the counterclockwise disk (CCWS). This supports the results of Bartko et al. (2010) who find a disk IMF skewed toward massive stars. Their disk KLF predicts roughly the same number of stars with m_K = 14 as with m_K = 17. A Chabrier/Kroupa IMF on the other hand predicts five times more m_K = 17 stars. Although the statistical significance of only five early-type stars is limited, it is intriguing that we find one m_K = 14 disk member (consistent with the average B-star density) but no fainter disk members. The faint B/A-stars exhibit an isotropic orientation (Bartko et al. 2010) and are most likely the remains of older starbursts.

Before computing the indices, each stellar spectrum was shifted to rest wavelength. We then divided the spectrum by a second-order polynomial fit to remove the curvature of the spectrum. For the continuum fit, we excluded regions with significant absorption lines; the polynomial fit is necessary to account for the large (A_K ∼ 3) and spatially variable extinction in the GC. We then computed the BL03 index according to the recipe of Blum et al. (2003). The FR01 index was computed in a similar way, with the bandpasses as described in Frogel et al. (2001). The continuum level w_C(λ) is estimated with a linear fit to the intervals stated in Table 1. The equivalent width is measured according to

$$\begin{eqnarray} &&{\rm EW(CO)}=\int _{{\rm band}} \left(\frac{w_{\rm C} - w_{{\rm line}}}{w_{\rm C}}\right)d\lambda. \end{eqnarray} \tag{ 1 }$$

We tested both indices (BL03, FR01) with template spectra of different resolution. We also reddened the template spectra artificially to determine the impact of extinction.

Reducing the resolution from R ∼ 3000 to R ∼ 2000 decreases the BL03 CO index by a factor of ≈0.92. The artificial reddening and the corresponding change of curvature of the spectrum cause a reduction of ≈0.94. Both effects cause the BL03 CO index to be underestimated by a factor between 0.90 and 0.85. The systematic underestimation of the BL03 CO strength leads to an overestimation of the stellar temperatures by ∼200 K. The FR01 index decreased by less than a factor 0.98 due to extinction. Degrading the resolution from R ∼ 3000 to R ∼ 2000 shows no measurable impact. We estimate the combined systematic effect to be less than 0.97 (50 K). The BL03 index suffers also from contamination of the Mg i line contained in the continuum bandpass of the index. The average Mg i line strength of the GC giants is similar to the calibration giants. Thus, the continuum estimation is not biased. Yet the line shows some intrinsic scatter introducing a statistical error in the CO index computation. To minimize the statistical and systematic error sources, we therefore adopted the index definition of FR01. We have to note that the results of Blum et al. (2003) did not suffer from the resolution dependence of their CO index because the bulk of their GC spectra were of the same resolution as the comparison stars.

The systematic errors in the derivation of the temperature are the main drivers in the age uncertainty of a giant population (at a given metallicity). As discussed in the previous section, the CO index computation might be biased on a ∼50 K level. More important, however, is the theoretical uncertainty. The Padua and Geneva Isochrones can differ by up to 80 K for the same star. Strictly speaking, this is not an observational uncertainty but in the fitting procedure the theoretical uncertainty must be taken into account. We included the bias by assuming a total systematic uncertainty of 100 K. To visualize the impact of temperature uncertainties, Figure 7 shows the theoretical age versus median temperature relation for red clump (RC) stars (0 < M_bol < 1). Especially for old ages, the age interpretation is very sensitive to temperature changes. Small temperature biases of the order 100 K might change the derived age by several Gyrs. In the star formation history calculation (see Section 5), we did not use only RC stars. However, RC stars make up more than one-third of all observed giants. Thus, the systematic uncertainty of the RC population is representative of the whole giant population.

Zoom In Zoom Out Reset image size

The observed K-band magnitude and the spectral CO index allowed the determination of the bolometric magnitude and temperature for individual GC stars. The bolometric magnitude M_bol follows directly from the intrinsic, de-reddened M_K. To derive the intrinsic M_K we used a distance modulus of 14.6 (8.3 kpc). The extinction was calculated for each star individually, following Bartko et al. (2010). We used for each star the 20 nearest neighbors to derive the median H − K color. By assuming an intrinsic color of Δm_{H − K} = 0.1 (typical for K giants) we derived the reddening. By assuming a power-law extinction A_λ ∼ λ^−2.21 (Schödel et al. 2010; Fritz et al. 2011) we computed the A_K extinction for each star. The bolometric correction (BC_K) was taken from Blum et al. (2003) (BC_K = 2.6 − (T_eff − 3800)/1500). The assumed BC_K is consistent with the bolometric correction library from Lejeune et al. (1997) used in the subsequent population synthesis models. The bolometric 1σ uncertainty is 0.32 mag. It is caused by the photometric error and the uncertainties in temperature and extinction. The stellar T_eff follows directly from the CO strength. The statistical 1σ T_eff uncertainty derived by error propagation of the measured CO index uncertainty is 200 K. The systematic T_eff uncertainty is 100 K (see Section 3.4.1). The RC of the GC giants can be found at a luminosity of M_bol = 0.6 and a temperature of T_eff = 4310 K (see Figure 10). Since the RC is narrow, both in magnitude and temperature, i.e., all stars have very similar spectral features, it is a viable approach to construct a median spectrum of the GC RC stars. In order to construct a high S/N spectrum, we used 234 RC stars with bolometric magnitudes 0 < M_bol < 1 corresponding to 15.1 < m_K < 16.1. The average S/N of individual spectra is between 10 and 15. All spectra were normalized and shifted to rest wavelength as described in the previous section. Most of the spectra have a resolution of R ≈ 2000. Spectra with a higher resolution were interpolated to match the latter. The equivalent integration time of the combined spectrum is of the order 300 hr. Figure 8 shows the resulting median spectrum of the GC giants.

Zoom In Zoom Out Reset image size

We compared the median spectrum with available library stars. The template star matching the co-added spectrum most closely is a K3 III giant with a temperature of 4330 K. The agreement of the spectra is impressive. Only the sharpest peaks are slightly smeared out due to a residual velocity error. The $\rm ^{12}CO$ bandheads at 2.294, 2.322, 2.352, and 2.383 μm are well represented. Even the weaker bandheads of $\rm ^{13}CO$ at 2.345 and 2.373 μm match the template (Figure 8). The comparison with colder and warmer giants (Figure 9) shows the sensitivity of the CO bandheads to the stellar temperature. The median temperature of the RC as derived by the stacked spectrum, T_eff = 4330 K, agrees very well with the median value (T_eff = 4310 K) of the individual low S/N spectra. The Ca i and Na i line blends are moderately sensitive to temperature as well as to the surface gravity log g. The Ca i lines are in good agreement with the template. The same is true for the iron line blends at 2.22 and 2.23 μm as well as Mg i, Si i, and Al i. Unfortunately, almost all of the atomic lines are contaminated with weaker atomic lines and lines of CN. This excludes the possibility of a spectral synthesis fit at the given resolution. However, qualitatively, it is clear that the GC spectrum is closely matched by a solar metallicity spectrum, in agreement with the work of Cunha et al. (2007).

Zoom In Zoom Out Reset image size

The strongest deviation between our average spectrum and the K3 III template is found in the Na i lines. They seem to be intrinsically stronger than in the solar neighborhood. None of the template spectra in the temperature range of the RC can reproduce the Na i strength. An increased Na i strength was reported previously in low-resolution spectra by Blum et al. (1996). The Na i lines are actually blends of a couple of atomic lines and CN lines. The individual contribution depends on the stellar temperature. For temperatures similar to a K3 III giant, sodium is the most important line. Cooler spectra are heavily influenced by Sc and the coolest spectra are dominated by CN. For a detailed analysis of lines in K-band spectra see Wallace & Hinkle (1996). Carr et al. (2000) used high-resolution spectra of IRS 7 to derive abundances. They found that the increased line strength of the Na i complex is mainly caused by stronger CN lines. The CN lines reflect extreme CNO abundances in the atmosphere of IRS 7, which they claim is probably the result of increased rotational mixing.

Cunha et al. (2007) also find CN-cycled material in the atmospheres of three luminous GC giants, although not as deeply mixed as in IRS 7. Increased rotational mixing is predicted for dense stellar clusters, where tidal spin-up can lead to significantly higher rotation speeds of main-sequence stars (Alexander 2005). This might explain the increased CNO material in the outer atmosphere layers of the luminous GC giants. Evidence for increased rotational mixing taking place in a supergiant like IRS 7 with a mass of 20 M_☉ and a lifetime of a few Myrs are hard to transfer to the RC giants of the GC with masses of only 1–2 M_☉ and ages of Gyrs. This is especially true since IRS 7 is significantly cooler (3600 K) than the RC.

However, Maeder & Meynet (2000) note that RGs with masses M  <  1.5 M_☉ are susceptible to extra mixing and Alexander (2005) notes that tidal spin-up is most effective in long-lived low-mass stars. Thus, the strong sodium lines might indicate that the RC stars have undergone a mixing process different from the solar neighborhood, for example due to tidal spin-up. In principle, the sodium line strength can also reflect a peculiar chemical composition of the GC. With the current knowledge we cannot rule out that possibility. However, rotational mixing provides a convincing explanation for the increased sodium strength. This might provide interesting conclusions on the stellar evolution since fast rotating stars tend to be more luminous and redder and can provide evidence for the existence of an underlying dense stellar cusp of low-mass main-sequence stars (Alexander 2005).

Apart from the Na i lines, the average spectrum shows no peculiarities. Small deviations of the GC spectrum at 2.219 and 2.349 μm are caused by poorly subtracted nebular emission of the Mini-Spiral. The deviation at 2.317 μm on the other hand coincides with a telluric line. Overall the agreement is intriguing and validates the method of constructing a median spectrum a posteriori.

The coolest temperatures we find cannot be reproduced by even the oldest solar metallicity isochrones. Those outliers might be explained by a population of metal-rich stars (Z > 2.5 Z_☉). However, this does not contradict the current assumption of a near-solar metallicity in the GC. The cold outliers account for less than 8% of the total population. Since metallicity studies in the GC (Ramírez et al. 2000; Cunha et al. 2007) have probed not more than 10 late-type stars, a small metal-rich population might have gone undetected. This can be compared to the situation in the bulge (Baade's window), where a high-metallicity tail of stars (Z > 2.5 Z_☉) makes up about 15% of the population (Zoccali et al. 2008). Another explanation might involve stellar model uncertainties. Most of the outliers show the same luminosity as bright and cool AGB stars. Stellar models of these stars still suffer from large uncertainties. Model isochrones treat this evolutionary phase in a simplified manner and are thus rather unreliable. Furthermore, stars in these stages are known to pulsate with periods of hundreds of days. During the pulsation, an individual star can change its temperature by up to 500 K (Lancon & Mouhcine 2002). All systematic effects considered by us lead to an overestimation rather than an underestimation of the stellar temperature. Yet some of the stars can be heavily dust obscured and thus appear too faint for the given temperature. In the following fitting procedure we ignored stars that were cooler than allowed by the isochrones and the temperature uncertainty. This excludes only a small number of the old stars. Therefore we are confident that the impact on the results is negligible.

Roughly 10% of the observed GC giants with magnitudes between 0 < M_bol < −4 appear to belong to a branch of young (<500 Myr) giants. With masses between 2.5 M_☉ < M < 6 M_☉ those stars are descendants of main-sequence B-stars. They are tracers of an intermediate-age population in the GC. Their stellar age is smaller than the typical non-resonant two-body relaxation time in the GC of the order t_nr ≳ few Gyrs in the GC (Alexander 2005; Merritt 2010). However, close to the SMBH, orbits are near-Keplerian. This causes interactions between stars to build up coherently. The randomization of the angular momentum vectors happens therefore on the fast vector resonant relaxation timescale t_vr < few tens of Myr (Hopman & Alexander 2006).

However, the eccentricity and semi-major axis distribution are randomized on the slow scalar resonant timescale t_sr (Hopman & Alexander 2006), comparable to t_nr for r > 0.1 pc. The mean stellar mass of the young giants (3.2 M_☉) is significantly larger than the average mass of the old giants (1 M_☉). The more massive stars tend to sink to the center of a cluster due to dynamical friction exerted through the drag of lighter background objects. The mass segregation timescale in the GC is larger than in normal stellar clusters due to the presence of the SMBH and the corresponding higher velocity dispersion. The mass segregation timescale $t_{\rm s}\sim t_{\rm nr} \frac{\overline{M}}{M_{\star }}$ scales with the relaxation time t_nr but is also a function of the individual stellar mass M_⋆ and the mean stellar mass $\overline{M}$ (Alexander 2005). Assuming the mean stellar mass is $\overline{M} \approx 1\, M_{\odot }$, the segregation time is $t_{\rm s} \approx \frac{1}{3}t_{{\rm nr}} > 1\,\rm Gyr$. This is still significantly larger than the age of the young giants. Thus the radial distribution could not have changed significantly since the stars formed (neither through relaxation nor through mass segregation). Yet the angular momentum vectors (i.e., the orientation of the orbits) will have undergone randomization.

To assess the dynamical state of the young giants we computed their orbital distribution. We followed the method of Bartko et al. (2009), who used statistical arguments to infer the 3D distribution of a stellar population out of the 3D velocity and projected distance distribution. A coherent motion within a stellar system can be detected as a statistical preference for an angular momentum direction. However, the orbital distribution of the young giants shows no significant excess as would be the case for a disk or a streamer. The orientation of the angular momentum vectors is consistent with being isotropic, as expected due to the fast vector resonant relaxation. The radial distribution of the young giants compared to the cool giant population is shown in Figure 11. Both populations exhibit the same radial density distribution. The density distribution of the total giant population has been described as a core or even a hole close to the SMBH (Buchholz et al. 2009; Bartko et al. 2010; Schödel et al. 2010). In any case, the distribution differs significantly from a Bahcall–Wolf cusp, the predicted final state of a relaxed population. Given the age of the young giants, they must have formed rather close to the SMBH, either in situ or transported in by in-falling clusters. Since the radial density distribution evolves only slowly on timescales of ∼ a few Gyrs this means that the young giants still contain information on their initial distribution. The 3D velocity of the young giants compared with the cool giants in the same magnitude bin is shown in Figure 12. The 3D velocity was computed assuming a distance of 8.3 kpc to the GC. The maximum 3D velocity $v_{\rm max} = \sqrt{\vphantom{A^A}\smash{\hbox{${\frac{2GM_{\bullet }}{r}}$}}}$ for stars to be bound is indicated in the figure. The projected distances r yield a lower limit for the physical 3D distance R. The kinematics of the young population are consistent with the old stars. Using a Kolmogorov–Smirnov test to asses the likelihood that the young giants' 3D velocities can be drawn from the cold population returns a probability of 70%. The fact that both giant populations share the same radial distribution is surprising. The young giants (M_ZAMS ≈ 3 M_☉) are three times more massive than their old counterparts with a lifetime too short to experience relaxation and especially to change their angular momentum significantly. However, it renders a radially varying IMF unlikely. The latter is a valid claim although star formation history and IMF are highly degenerate, since one effect would need to cancel the other exactly to emulate a radially constant distribution of young versus old giants.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To assess the fit uncertainty we used a method generally referred to as bootstrapping. We constructed 1000 H-R diagrams by drawing random stars out of the data. Each star was allowed to be drawn any number of times. To include the systematic uncertainty of the data and the theoretical isochrones, we added a random temperature offset with a Gaussian σ of 100 K to the whole data set. The constructed H-R diagrams were fitted again with the model populations. The scatter of the derived relative SFRs represents the 1σ uncertainty of the relative SFR (see Table 3). Note the SFR uncertainty is mainly driven by the systematic temperature uncertainty. To judge the quality of each model fit, we produced 1000 Monte Carlo data sets, each containing 450 stars drawn randomly from the respective model (not the original data). Calculating the variance in each bin resulted in a variance diagram. We defined the fit quality as $\chi ^2=\sum _i \frac{(m_i-n_i)^2}{\sigma _i^2}$, where σ²_i is the variance in bin i, m_i is again the number of model stars, and n_i is the observed number of stars in bin i. The values for χ², the number of degrees of freedom, and the corresponding probability that the data can be drawn from the model population are represented in Table 3. The numbers of degrees of freedom are equal to the number of populated bins minus the number of free parameters. Figure 13 illustrates the residuals for various models. Each panel shows the residual model—data weighted by the variance (Dolphin et al. 2002). The color coding indicates bins where the model predicts more stars than observed (bright) and where the model underpredicts the number of stars (dark). The first panel of Figure 13 shows the best-fit model, while the second panel shows a continuous star formation model (both assume a Chabrier IMF). The continuous model predicts too few old (i.e., cold) stars compared to the observations, while it predicts too many young (i.e., warm) stars. The data are well fitted by a mostly old (>5 Gyr) population with an admixture of recently formed stars (< a few 100 Myr) as in the case of the best-fit model. The discrepancy between a continuous formation scenario and a time-varying star formation increases in the case of IMFs favoring the formation of massive stars (right two panels).

Zoom In Zoom Out Reset image size

The two confirmed A-stars and the one candidate (Section 3.1.1) are ideal tracers for star formation during the last few hundred Myrs. Early-type stars with K-band magnitudes 17 < m_K < 18 (assuming A_K = 2.8 and R₀ = 8.3 kpc) are main-sequence B9/A0 dwarfs with average lifetimes of 500 Myr. Late-type stars in the same magnitude range can only be giants that have already left the main sequence. These giants have ages between 1 and 12 Gyr. Thus the number count ratio of the two populations provides a measure of the star formation efficiency during the last 500 Myr relative to earlier star formation. This ratio is largely independent of systematic uncertainties since it avoids issues like incompleteness and spectroscopic detectability. We included the candidate star in the early-type population. In total we found three (two confirmed) early-type stars and 30 late-type stars in this magnitude bin. We used the IAC-STAR code and found that in the case of continuous star formation over 12 Gyr with a Kroupa IMF, the predicted early-to-late ratio is close to unity. The observed ratio therefore argues for an average SFR during the last 500 Myr of only about 10% ± 6% of the average formation rate 1–12 Gyr ago. To convert the relative formation rate into an absolute rate, we used the SFR inferred in the previous section and calculated an average rate of 2.3 ± 0.5 × 10⁻⁴ M_☉ between 1 and 12 Gyr look-back time. Using this value, we obtained an average SFR of 2.3 ± 2.1 × 10⁻⁵ M_☉ during the last 500 Myr (see Figure 14). Within the errors this is consistent with the one derived by the H-R fitting. The A-star ratio provides an independent measurement of the relative SFRs. It is another indicator for the early formation of the nuclear cluster.

Maness et al. (2007) found that the giant population of the GC is on average warm and thus young. Consequently, they favored models with a normal IMF and an increasing SFR or a top-heavy IMF and continuous formation. Maness et al. (2007) used a CO index that is very sensitive to systematic effects. In particular, their CO index, though widely used, was recently discovered to vary with spectral resolution (Mármol-Queraltó et al. 2008). Since no suitable library was publicly available at the time of writing, they also used template libraries with resolutions between R ∼ 3000 and 5000 to calibrate their R ∼ 2000 data. The sum of the effects caused an underestimation of the CO equivalent width and a resulting overestimation of the stellar temperatures. This led to the interpretation of a mostly young giant population. Applying the same methods as Maness et al. (2007) together with a CO index that is less susceptible to systematic effects and using the newly available library of Rayner et al. (2009) with R ∼ 2000 allowed us to revisit the star formation history of the GC.

Various models have been fitted to the data (Section 5.1). Each model yields the SFR as function of time under the assumption of a certain IMF. To derive the absolute mass contribution, it is necessary to scale the models according to the actual star counts. To get a representative number for the nuclear cluster we used the stellar surface density from Schödel et al. (2007). As discussed in Section 1 the radial extent of the NSC is ∼5 pc. Our data, however, probed only the inner 30'' (1.2 pc). Therefore we restricted ourselves to that radius. By assuming 30'' as a sharp edge, the surface density from Schödel et al. (2007, their Figure 12) yields ∼18, 200 stars with m_K < 17.75 within a projected radius of 30'' from Sgr A*. Between 45% and 55% of the projected stars are also contained within a 3D distance of 30'', depending on the radial density profile derived by Schödel et al. (2007). Thus, we assumed a total of 9100 stars with R_3D < 1.2 pc to scale the remnant, stellar, and total gas mass for each model. Table 3 shows the derived SFR for each model IMF. Figure 16 shows the mass composition for the models derived in Section 5.1. The model predictions have to be compared with the dynamical mass estimates of the nuclear cluster (see Figure 16). The total mass enclosed within 30'' is 5.7 ± 0.9 × 10⁶ M_☉. The SMBH contributes 4.3 ± 0.5 × 10⁶ M_☉ (Ghez et al. 2008; Gillessen et al. 2009), while the remaining 1.4 ± 0.7 × 10⁶ M_☉ (Genzel et al. 1996, 2010; Trippe et al. 2008; Schödel et al. 2009) are contained in stars and stellar remnants. Numerical simulations of Freitag et al. (2006) have shown that during 10 Gyr about 4 × 10⁵ M_☉ stellar and 1 × 10⁵ M_☉ remnant mass is removed from the cluster by the SMBH through tidal disruption and inspiral. The cluster composition is scaled to match the observed number of giants. Thus, the inferred stellar mass intrinsically accounts for stars lost to the SMBH because giants and giant progenitors are equally likely to be disrupted as unresolved main-sequence stars constituting the bulk of stellar mass. The situation for remnants, however, is different. The main remnant contribution comes from stellar BHs dominating the mass density in the inner 0.1 pc. Due to their mass (∼ 10 M_☉) they are significantly affected by mass segregation and more likely to inspiral into the SMBH. Consequently, the inspiral of stellar BHs has to be taken into account in the mass budget. We simply added 10⁵ M_☉ with an uncertainty factor two to the observed dynamical mass to constrain the IMF. We took the value determined by Freitag et al. (2006) as being constant with IMF. This might be the weak point in the line of argument, since Freitag et al. (2006) assumed a Chabrier/Kroupa IMF in their simulations. As Figure 16 shows, the cluster composition depends sensitively on the assumed IMF. For flatter IMFs, the number of stellar BHs increases, but the mean mass increases accordingly making mass segregation less efficient. Thus the actual mass transfer between the cluster and the SMBH as function of IMF can only be addressed by further simulations. Keeping the limitations in mind, the dynamical mass plus the removed mass therefore make IMFs with a slope flatter than α = −1.1 unlikely. Even if the transferred mass is significantly higher, models flatter than α = −0.8 violate the total enclosed mass (SMBH + cluster). Mechanisms that expel remnants or stars are very inefficient. One mechanism is the evaporation of stars from the nuclear cluster. However, for the GC the timescale is greater than a Hubble time (Alexander 2005). Mass segregation is another mechanism for removing stars. This mechanism moves low-mass constituents outward, and high-mass constituents inward, without changing the radial density profile significantly. Miralda-Escoudé & Gould (2000) suggest that inward migration of a considerable number of bulge stellar BHs can remove low-mass stars from the nuclear cluster. According to their model, the low-mass stars are supposed to form a core with a radius of 1–2 pc. Observations, however, find a core radius of about 0.3 pc (Schödel et al. 2007). If the observed core is related to mass segregation, then the effect is smaller than predicted. Our constraints on the IMF are however not weakened by potential mass segregation. Our models use the number of observed stars as a scaling for the total processed mass (and remnants produced). Therefore if low-mass stars are removed from the inner region, our scaling underestimates the remnant mass. The inferred stellar mass, however, is not changed because giants and unresolved main-sequence stars have very similar masses (∼1 M_☉). Thus the total predicted mass will grow if we take into account that low-mass stars are removed due to mass segregation. This makes our statement on the IMF constraints even more significant. In general, it is very hard to remove mass in the form of stars and remnants from a nuclear cluster (an SMBH binary, however, is able to remove stars efficiently). The processed gas mass, unlike stars and remnants, can be expelled from the nuclear cluster due to supernovae or active galactic nucleus activity in the distant past of the SMBH. Stellar winds are also able to remove gas from the cluster. While the low-mass main-sequence stars and giants have wind speeds significantly lower (v_wind ∼ a few tens of km s⁻¹) than the escape velocity of the NSC the most massive stars (v_wind ∼ 1000 km s⁻¹) can efficiently remove gas from the cluster (McLaughlin et al. 2006; Martins et al. 2007). Therefore, the processed gas mass cannot constrain the IMF although flat IMFs require a factor hundred greater gas masses than Kroupa-like IMFs. Summarizing the arguments, we can say that IMFs flatter than α > −1.1 violate the observed cluster mass. IMFs flatter than α > −0.8 exceed even the total mass (SMBH + cluster).

Zoom In Zoom Out Reset image size

As discussed in Section 2.1.1, depending on the IMF and star formation history the diffuse background light from faint unresolved stars may probe the bulk of the stellar population. We find that the diffuse background contributes H_diff/H = 27% ± 9% to the total H-band flux. Furthermore, we derived a mass-to-diffuse-light ratio of M_dyn/H_diff = 2.6 ± 1.5 M_☉/L_{H, ☉}. In the following we used the inverse ratio H_diff/M_dyn for convenience. We compared both quantities to our best-fit models (Section 5.1). To show how H_diff/H and H_diff/M depend on the IMF slope and overall star formation history, we used the population synthesis code STARS (Sternberg 1998) to compute these ratios for a range of IMFs and star formation timescales t₀ for simple histories ${\rm SFR}(t) \propto e^{-t/t_0}$. We considered a Kroupa IMF, and also three power-law IMFs, dN/dm∝m^−α, with α equal to −1.5, −1.35, and −0.85. All of the IMFs range from 0.01 to 120 M_☉. We used Geneva evolutionary tracks for solar metallicity stars, combined with empirical colors and bolometric corrections for dwarfs, giants, and supergiant stars. For low-mass stars (<0.8 M_☉), we computed the H-band luminosities along the lower main sequence using the calibrations of Henry & McCarthy (1993). We considered exponentially decaying (t₀ = 3 Gyr), continuous (t₀ = ∞), and exponentially increasing (t₀ = −3 Gyr) SFRs, for an assumed cluster age t = 13 Gyr, and fixed dynamical mass M_dyn = 1.5 × 10⁶ M_☉. The results are displayed in Figure 17 in the H_diff/M versus H_diff/H plane. The gray areas indicate the measurement including the 1σ uncertainty. Figure 17 shows that H_diff/M and H_diff/H both decrease for flatter IMFs. This behavior is due to the relative increase in remnant mass from massive stars, and the reduction in the relative fraction of diffuse light from low-mass stars. For a given IMF, H_diff/M decreases but H_diff/H increases, as the history is altered from increasing, to steady, to declining star formation, because the total luminosity for fixed mass is reduced for this sequence of histories, while the fraction of light produced by the accumulating long-lived low-mass stars is increased. In Figure 17 we also indicate the positions of the "best-fit" models discussed in Section 5.1 and listed in Table 3. Figure 17 shows that the observed H_diff/M and H_diff/H ratios require steep (canonical) IMFs and continuous or mildly decaying star formation. In particular, flat IMFs with α > −1.5 are inconsistent with the observations.

Zoom In Zoom Out Reset image size

Our measurements of the diffuse H-band background can be compared to the findings of Löckmann et al. (2010), who used the diffuse K-band background together with the dynamical mass to derive M/K_diff = 1.4^+1.4 _{− 0.7} M_☉/L_{☉, K}, which was then compared with model predictions. Their mass-to-diffuse-light ratio is somewhat lower than our value (M/K_diff = 1.9 ± 1.2 M_☉/L_{☉, K}), although consistent within the errors. The difference originates from the assumed extinction value. Löckmann et al. (2010) used A_K = 3.3 published by Buchholz et al. (2009). However, this value is 0.5 mag (factor 1.6) higher than the most recent one published by Schödel et al. (2010) and Fritz et al. (2011). Correcting for this difference removes the discrepancy between their finding and ours. Their analysis favored an IMF steeper than α = −1.3 together with a continuous SFR or an increasing SFR. Our analysis confirms their finding of a steep IMF. Taking into account the revised extinction our finding of an early star formation is also consistent with the findings of Löckmann et al. (2010). However, we additionally used the ratio of diffuse light to total light to constrain the star formation history even further.